This check operator is called main, and it is for construction of this operator from the spectral curve “bundle” over the moduli space, that the AMM/EO recursion procedure was later devised [1]}, [2]}, [3]}, [4]}. It follows from the previous subsection that the main check-operator \(\check{\nabla }_z\) is expressed through \(y\) , its derivatives and \(W^{\prime }(x)\) . It is important to notice that \([\hat{\nabla }_{z_1}, \hat{\nabla }_{z_2}]=0\) , but \([\check{\nabla }_{z_1},\check{\nabla }_{z_2}]\ne 0\) . Hence, these operators are of different level of complexity, but the check-operator acts on a much smaller space. Unfortunately, many properties of the check-operator have not been well-studied yet, though some of them are already known.
